Tensors are usually indicated using the abstract index conventions (see [1]} or [2]}). Indices are labels and do not indicate a choice of frame. An index is in either up or down position. Up (down) indices label contravariant (covariant) tensors. For example, \(\nabla _{i}\nabla _{j}X^{k}\)  (which could also be written \((\nabla ^{2}X)_{ij}\,^{k}\) ) indicates the second covariant derivative of the vector field \(X^{k}\) . Were a frame \(E_{i}\)  chosen, \(\nabla _{i}\nabla _{j}X^{k}\)  would correspond to \((\nabla ^{2}X)(E_{i}, E_{j}) = \nabla _{E_{i}}\nabla _{E_{j}}X - \nabla _{\nabla _{E_{i}}E_{j}}X\) , and not to \(\nabla _{E_{i}}\nabla _{E_{j}}X\) . The up index \(k\)  on \(X^{k}\)  is simply a label that indicates that \(X\)  is a contravariant vector field. The summation convention is used in the following form: repetition of a label in up and down position indicates the trace pairing. Enclosure of indices in square brackets (parentheses) indicates complete antisymmetrization (symmetrization) over the enclosed indices, so that, for example, \(a^{ij}= a^{(ij)} + a^{[ij]}\)  indicates the decomposition of a contravariant two-tensor into its symmetric and skew-symmetric parts. An index included between vertical bars \(|\,|\)  is omitted from an indicated (anti)symmetrization; for example \(2a_{[i|jk|l]} = a_{ijkl} - a_{ljki}\) .
